Great Sam Shah Action


Sam Shah’s blog has been a veritable teaching clinic the last two weeks, more than filling his own installment of Great Classroom Action.

With Attacks and Counterattacks, Sam asked his students to define common shapes as best as they could — triangle, polygon, and circle, for instance. They traded definitions with each other and tried to poke holes in those definitions.

When the counter-attacks were presented, it was interesting how the discussions unfolded. The original group often wanted to defend their definition, and state why the counter-attack was incorrect.

Trade the definitions back, strengthen them, and repeat.

Sam created some very useful scaffolds for the very CCSS-y question, “If you have a shape and its image under a rotation, how can you quickly and easily find its center of rotation?”

This is an awesome exercise (inmyhumbleopinion) because it has kids use patty paper, it has them kinesthetically see the rotation, and it gives them immediate feedback on whether the point they thought was the center of rotation truly is the center of rotation. Simple, sweet, forces some thought.

Sam then pulls a move with a Post-It note that is a stunner, simultaneously useful for clarifying the concept of a variable and for finding the sum of recursive fractions:

Ready? READY? Flip. THAT FLIP IS THE COOLEST THING EVER FOR A MATH TEACHER. That flip was the single thing that made me want to blog about this.

Finally, Sam pulls a masterful move in the setup to his students’ realization that all the perpendicular bisectors of a triangle’s side meet in the same point. He has them first find those lines for pentagons (nothing special revealed) and quadrilaterals (nothing special revealed) before asking them to find them for triangles (something very special revealed).

There were gasps, and one student said, and I quite, “MIND BLOWN.”

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I really like all of these activities.

    Reading the first reminded me a little bit of the section in Ed Frenkel’s book “Love and Math” where he describes the discrimination he faced in his college entrance interviews. One of the questions was to define a circle.

    For the second piece, I’ve been thinking a little bit about how to help use any sort of manipulative to see / feel math better. Love the phrase “kinesthetically see the rotation.”

    This was my attempt from last weekend to use our Zometool set to understand some 3D geometry and rotations a little better:

    Funny enough, Fawn Nguyen was involved in a sort of similar zometool activity at almost the same time:

    For the continued fraction activity – all I can say is YES!! I think continued fractions are great examples for kids. We’ve played with them several different ways and always seem to have a lot of fun. This was one of our projects relating to geometry and the square root of 2:

    Some fun questions to ask about the 1 + 1/x = x equation in the blog is – why are there two solutions? What does the other solution represent?

    Also, there’s a really great section in Jordan Ellenberg’s book “How not to be Wrong” about what he calls “algebraic intimidation.” Ellenberg’s idea does have some application here when you think about why this algebra works to simplify the continued fraction. We did a project where some very similar looking algebraic ideas fail here:

    The last piece on the perpendicular bisectors is really a beautiful activity – especially the last part where he says “In fact, the fact that they didn’t discover it on it’s own was so powerful when they ended up seeing it.” This statement probably isn’t true for all activities, but when it is, like for his activity, it really is powerful.

    One of my favorite activities – just purely for the reaction when the kids see it – is here: